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The everyday world for a young child is full of opportunities to engage with number and quantity. From the first few days of life, infants are naturally attuned to numerical quantities. This early predisposition to quantity provides the foundation for more sophisticated understandings of mathematics that develop later in life. Although young children are capable of impressive mathematical thought, as they get older their numeracy development becomes increasingly dependent on the types of exposure they receive from parents and caregivers and the nature of the interactions in which they participate in their daily lives (Baroody, Lai, & Mix, 2006; Carpenter et al., 1993; Klibanoff et al., 2006). It is therefore important for caregivers to be aware of the mathematical potential of young children and to structure the environment to maximize its development.

Numerical Aptitude (0 – 12 Months)

In the first year of life, infants develop foundational abilities for later numeracy development. These foundational abilities include discrimination of same and different amounts of discrete and continuous quantities.

Discrimination Abilities

Numerical aptitude is present in very young infants. Considerable research evidence suggests that babies between five and seven months are capable of discriminating between small collections of objects; for example, they are able to distinguish a set of two dots from a set of three (Feigenson, 2005; see Wynn, Bloom, & Chiang, 2002). One study concluded that even newborn infants are able to distinguish between small sets that differ in number (Antell & Keating, 1983). In addition to infants’ discriminatory abilities of static displays, researchers have extended the research to include dynamic displays of numerosities (VanLoosbroek & Smitsman, 1990; Wynn, 1996). For example, Wynn (1996) used a puppet to display a certain number of jumps; she concluded that infants at six months can discriminate between two and three jumps. Together, these results have allowed researchers to conclude that infants are capable of subitizing up to three objects (Antell & Keating, 1983; Starsky & Cooper, 1980). That is, they are able to quickly see “how many” for sets of two and three, even though they are too young to actually count the objects in the sets.

As they get older, infants are able to discriminate between larger sets of items, but only if one set is at least twice as large as the other. For example, Xu and Spelke (2000) found that six-month-old infants were able to discriminate between a set of 8 and 12 items (Brannon, Abbott, & Lutz, 2004; Lipton & Spelke, 2003; Wood & Spelke, 2005a, b; Xu, 2003; Xu, Spelke, & Goddard, 2005). Between nine and twelve months, still before conventional counting skills emerge, infants’ ability to discriminate between two larger sets of objects improves, even if the ratio between the two sets is less than 1 to 2. By nine months, for example, infants can distinguish between sets of four and six objects and also between eight and twelve objects (Lipton & Spelke, 2003; Wood & Spelke, 2005b).

Discriminating Between Continuous Quantities

Aptitude for quantity is not limited to discrete quantities, however. Around five months of age, infants can also demonstrate the ability to discriminate between continuous quantities, such as between a cylindrical container that is either half-full or three-quarters full of a colored liquid (Gao, Levine, & Huttenlocher, 2000).

Numerical Aptitude (13 – 36 Months)

As infants become toddlers, their understanding of quantity and numerosity continues to develop in increasingly sophisticated ways. Counting skills and concepts emerge, which provide the foundation for basic arithmetical reasoning.

Ordinal Relations

At fourteen to eighteen months, toddlers begin to attend to ordinal relations. In other words, it is at this age that children begin to go beyond mere discrimination of different quantities: At fourteen months, they can identify which of two sets contains more (Cooper, 1984; Curtis & Strauss, 1982; Curtis & Strauss, 1983; Strauss & Curtis, 1984; see also Brannon, 2002). This knowledge is not believed to be based on conventional number knowledge, such as counting, but rather is based primarily on perception and subitizing abilities. In other words, children rely on appearances when reasoning about quantities, such as choosing the set that that “looks like” it contains more (Clements, 2004; Cooper, 1984).

Counting and Arithmetic

The foundation for arithmetic begins very early. At five months, for example, infants realize that something is amiss when one object added to one other object results in the wrong number of objects, say three objects instead of two. Such anomalous displays result from manipulations of objects behind a screen (Wynn, 1992.) Some evidence suggests that close to their second birthday, or shortly thereafter, children become more aware of the effects of adding to or taking away from small quantities. For example, between two and three years many children expect that when one item is removed from a set of two, one item should remain (Clements, 2004; Huttenlocher, Jordon, & Levine, 1994).

Around the age of two, children also begin to display rudimentary counting skills. More specifically, children generally learn their first number words, usually “one” and “two” (Baroody, 2004; Clements, 2004) and their behaviour indicates that they know that number words are important (Ginsburg, 1989). For example, children between one and two start labeling their toys with number words, even though they do not yet understand the mathematical significance of these words (Fuson 1988).

Later in this age range, toddlers refine their ability to count and acquire rudimentary understandings of the concepts that underlie counting. Children first learn the counting sequence by imitating adults (Baroody, 1987; Ginsburg, 1989), but they often leave out numbers in the sequence. For example, a child could recite the incorrect sequence, “one, two, five, seven.” Although children frequently begin counting by using incorrect number sequences, they often obey the principle that number words must always be said in the same order (Baroody & Price, 1983; Fuson, 1988, Gelman & Gallistel, 1978). Furthermore, children as young as two-and-a- half years can accurately point to each item in a collection only once when enumerating (Gelman & Gallistel, 1978), indicating that they possess at least implicit knowledge that each item should be counted once and only once. Eventually and with appropriate practice, children between two and three can learn to recite the number sequence correctly from 1 to 10 (Baroody, 2004), and many can accurately enumerate sets of one to four items (Clements, 2004).

Children at this age also can learn to name the next number word after any number below 10, although most often they need a bit of help. For example, to answer the question, “what number comes right after 5?,” most children at this age need to be given a running start: “one, two, three, four, five,…” This indicates that at this age, children conceptualize the number sequence as a “singsong” that is not easily accessed at any point (Fuson, 1988).

Stronger Connections Between Counting and Quantity (37 – 48 Months)

More sophisticated connections between counting skills and understanding of quantity continue to develop during this age range. That is, some three-year-olds and almost all four-year-olds are able to assign quantity meaning to numbers – that is, they know that the last number recited when counting a set of objects indicates the size of the set (Clements, 2004; Fuson, Pergament, Lyons, & Hall, 1985). Around the age of four, children continue to make connections between counting and quantity – they learn that numbers said later in the counting sequence are larger than numbers that come before them in the sequence (Baroody, 1987). This connection between number and quantity is an important milestone, for it marks the time when children no longer perceive of numbers simply as words, but rather as representations of quantities that can be compared (Griffin, 2004). At the same time, children’s knowledge of the counting sequence expands; by the time they are four, many can count to 20 or 30 or higher, with emphasis on patterns between the decades (e.g., “21, 22, 23, … follows the same patterns as 1, 2, 3…”; Clements, 2004; Ginsburg, 1989). Nearing their fourth birthday, or shortly thereafter, children often can name the next number after any number below 10 without a running start (Clements, 2004; Baroody, 1987, 2004). Some children at this age also learn to count backward from the number 5 (Baroody, 2004).

Basic Arithmetic and Equal Partitioning

Between the ages of three and four years, children’s basic notions of arithmetic get refined as evidenced by their ability to predict the results of transformations in small sets (Griffin, 2004; Wynn, 1992). For example, by the time they are three-and-a-half, children can produce correct answers to small numerosity addition (e.g., 1 + 2, 1 + 3) and subtraction problems (e.g., 2 – 1, 3 – 2) with a degree of accuracy when the problems are presented nonverbally with objects (Huttenlocher et al., 1994). Around age three, children also begin to acquire basic notions of equal sharing, which provide the foundation for concepts related to multiplication and division. For example, three-year-olds are able to equally distribute a small collection of objects (under 10) between two people by using a partitioning strategy (“one for me, one for you”; Clements, 2004). When they get a little older (around four), children are able to determine how many objects each person gets by counting only one of the shares (Frydman & Bryant, 1988).

Engaging with Continuous Quantities

Children’s ability to engage with continuous quantities also improves at this age. For example, children know that when more of a substance, such as juice, is added to an existing quantity of juice, then it should look like there is more liquid (Gao, Levine, & Huttenlocker, 2000). Nearing their fourth birthday, children can measure length by directly comparing two objects placed side by side. For example, children will compare the length of a book to the length of their shoe by placing the items next to each other (Boulton-Lewis, Wilss, & Mutch, 1996; Clements & Stephan, 2004). Their vocabulary about measurement and quantity also expands; by the time they are four, and sometimes earlier, children use words such as “taller,” “shorter,” “skinnier,” “fatter,” “wider,” and “longer” when they talk about comparing objects (Clements, 2004; see also Lehrer, Jenkins, & Osana, 1998).

Counting and Enumeration Strategies (49 – 60 Months)

The skills and understandings that emerged in the fourth year of life continue to become refined and consolidated between the ages of four and six years. For example, children’s counting skills become more sophisticated. For instance, they can recite more of the counting sequence, sometimes to 100, by the time they are four-and-a-half (Baroody, 2004). By the age of five, children can learn to count up from numbers other than 1 (Siegler, 1987). Most children at four-and-a-half can learn to count backward from 5; as they get closer to their fifth birthday, they know how to count backward from 10, and by five-and-a-half, most learn how to count backward from 20 (Baroody, 2004; Clements, 2004). Around their fifth birthday, many children learn to skip count by 10s (Clements, 2004).

With respect to enumerating sets, most four-year-old children can count about 9 objects without error, most at five years can count a set of 20 objects accurately, and by the time most children are six, they can accurately count 28 objects or more (Baldwin & Stecher, 1925). In this time period children also learn to use more efficient counting strategies, such as pattern recognition. For example, many children in this age range are able to immediately recognize the pattern as representing 6, presumably from their experiences with dice and dominoes (Ginsburg, 1989; Griffin, 2007), but as with other areas of numeracy development, this recognition is dependent on the types of exposure received in the home and preschool environments (Baroody et al., 2006). Children who are five and six can learn to use basic arithmetic in conjunction with such strategies. For example, upon being asked how many dots were in the display , they will know to add 1 onto the 6 they immediately recognize to produce the number 7 (Ginsburg, 1989).

Number Representation

It is also during this time that children’s ability to identify and generate representations of number improves. For example, many children who are five learn to use finger patterns to represent numbers up to 10 and can write one-digit numerals (Clements, 2004). As they approach their sixth birthday, many children can write numbers in the teens; by six, they can learn to write all two-digit numbers (Clements, 2004). Finally, around the age of five-and-a-half or six, they can identify written one-digit number words (“one,” “two,” etc.) and know their corresponding numerals and cardinal values (Baroody, 2004).

Stronger Connections Between Counting and Quantity

During this period children also make stronger connections between their use of counting and their reasoning about global quantities. When they are given, for example, 12 raisins, they realize that someone must have counted these raisins to find the size of the set. Thus, when asked to place the 12 raisins in a box, the child would not need to recount the set. This concept of numbers, that they are at the same time counting words and representations of cardinality, usually emerges at some point between the ages of five and six. This is a key development because having this cardinal-count knowledge allows children to use a “counting on” strategy to solve simple addition problems. That is, when solving the problem “I have five dinosaurs and my sister gave me three more. How many dinosaurs do I have now?” a child with cardinal-count knowledge is able to start the count from 5 (the first addend) and count on 2 more to 7, the answer to the problem (Fuson & Hall, 1983; McLaughlin, 1935). This counting on strategy can be seen in five-year-olds (Siegler, 1996). In addition, by the time children are five, they recognize that once the task of counting a set is completed, the number words used do not adhere to the specific objects that were tagged; that is, they develop the foundation for the important principle that the order of counting does not affect the cardinality of the set, which emerges later in this time period (Baroody, 1992; Saxe, Guberman, & Gearhart, 1987).

Problem Solving and Place Value

By the time they are five years of age, most children can solve a variety of single-digit addition and subtraction word problems, most often with concrete supports, such as manipulatives (Clements, 2004; Osana & Chacko, 2006; Warfield & Yttri, 1999). In kindergarten, many, if not most, children can solve multiplication and division problems with the use of manipulatives to model the objects and actions in the problem (Carpenter et al., 1993). The types of concrete strategies used by children to solve such problems entail creating equal groups of items using manipulatives and counting the total number of items (in the case of multiplication) or counting the number of groups (for one type of division problem). Another strategy for division involves distributing the items fairly and then counting the number of items in each group. What is noteworthy is that even children around the age of six have an intuitive sense that the basis for multiplication and division is equal groups of items, which allows for efficient problem solving strategies, such as counting the number of items in one share only (Carpenter et al., 1999; Frydman & Bryant, 1988).

Related to the notion of equal groupings is the concept of place value, based on groups of 10. A focus on multiplication and division problems where the number of items in each group is 10 lays the foundation for an understanding of place value and the ability to use this knowledge to solve multidigit addition and subtraction problems (Carpenter et al., 1998). In fact, research has demonstrated that children between the ages of five and six are capable of understanding that a bundle of 10 popsicle sticks is the same quantity as 10 individual sticks, and later in this period, they can learn that a bundle of 18 popsicle sticks is the same as a bundle of 10 plus 8 individual sticks (Baroody, 2004). Furthermore, division with remainder problems allow for the development of fraction understanding. Research has demonstrated that between the ages of four and six, most children can learn to accurately label fractional shares as “one half” (Hunting & Davis, 1991).

With respect to problem solving in general, children during this time period develop more abstract (and thereby more efficient strategies) for solving problems and by the time they are five, some choose to reject the concrete methods used by younger children; it is at this age that children begin to use retrieved number facts to solve a variety of single-digit addition and subtraction problems (Carpenter et al., 1999; Clements, 2004; Siegler & Alibali, 2005). Researchers have found that between the ages of five and six, most children know their doubles facts to 10 (e.g., “2 + 2 = 4,” “3 + 3 = 6”; Carpenter et al., 1999; Clements, 2004).

Continuous Quantities

Finally, children’s reasoning about continuous quantities also improves between the ages of five and six. For example, five-year-old children know that the distance between two objects does not change unless the objects are moved (Miller & Baillargeon, 1990). In addition, children are able to compare attributes of objects by using tools, such as a string, to compare the length of two things, given that one of the objects is the same length as the string (Nunes, Light, & Mason, 1993). Children also develop the ability to use arbitrary units to measure attributes of objects. For example, between the ages of five and six, children can learn to place paperclips end-to-end to measure the length of a book, and later can learn to measure length using more conventional units, such as centimeters or meters (Clements, 2004).

Environments that Support the Mathematical Development of Young Children

The research described in this chapter indicates that young children are capable of thinking about mathematics in more sophisticated ways than previously thought (Baroody & Wilkins, 1999; Carpenter, Franke, & Levi, 2003; Ginsburg et al., 2006). Clearly, however, there are individual differences in children’s performance on such numeracy tasks and activities. One of the most critical conclusions that one can draw from the scientific literature is that these individual differences are due primarily to a lack of opportunity and exposure to numeracy-rich preschool environments (Baroody et al., 2006; Carpenter et al., 1993; Denton & West, 2002; Griffin, Case, & Siegler, 1994). Thus, while it is important to be aware of children’s capabilities, it is also necessary to know how to structure the preschool environment to maximize children’s growth in a variety of numeracy-related domains.

A general principle for creating productive numeracy environments for children is to start early. Children who are not immersed in environments that engage them with number and quantity are at a serious disadvantage when they begin school, but more importantly, such children experience a “downward spiral” with respect to mathematics that begins many years before they place their foot in a classroom (Arnold & Doctoroff, 2003; Duncan & Brooks-Gunn, 2000). It is therefore crucial to engage toddlers (and perhaps even younger children) in activities that serve to highlight the numerical relationships and patterns found in their everyday worlds. Asking children to estimate the number of buttons in the sewing box, for example, and talking about associated numerical relationships (how they compare to known quantities or others’ guesses, for example), is one way to connect number and quantity for the young child. Classifying objects into rows and columns, for example, or distributing snacks to a group of children (Baroody et al., 2006) provide other types of activities that can form the basis of several big ideas in mathematics, including multiplication and division. Other activities, such as figuring out how many more plates are needed on the table so that everyone gets one or determining the age of the child two years ago, form the foundation for other mathematical relationships.

An important tool for reasoning in such quantitative situations is counting. In fact, learning number words helps children to construct number sense (Baroody, 1992; Fuson, 1992). Activities that involve associating number words with patterns, such as those seen on dice or dominoes, are pedagogically important because they help children abstract notions of “twoness,” “threeness,” and so on (Baroody et al., 2006). In fact, engagement with board games and card games that require different representations of number is an important predictor of children’s mathematical development (Griffin, 2007).

Language is also an important tool in children’s numeracy development. It is vital to encourage children to talk about the quantities they are thinking about by using vocabulary related to number, counting, quantities, and comparisons. Using words such as “taller,” “skinnier,” “colder,” “lighter,” and so on, make it easier for children to map number onto the quantities they pay attention to in the world around them (Griffin, 2007). Communicating about mathematics with children and talking with them about their quantitative reasoning are essential for a number of other reasons. By listening to children talk about how they are thinking about number and quantity provides the adult with information about their intuitive knowledge, thereby allowing the adult to make more meaningful links between children’s conceptions and related ideas and representations in mathematics. Further, environments that are rich in mathematical language support children’s numeracy development. For example, Klibanoff et al. (2006) found that the amount of mathematical language used by early childhood educators was predictive of the development of mathematical understanding of preschool children in areas such as cardinality, calculation, and recognizing conventional number symbols.